On top local cohomology modules, Matlis duality and tensor products

Let $\mathfrak{a}$ be an ideal of a local ring $(R, \mathfrak{m})$ with $c = \mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in $R$. In the case that $c=\dim R$, we first give a bound for depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$, $D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and $H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. In the case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered.